<nettime> The Laying-Out (infinite abacus)

[Alan Sondheim <sondheim@panix.com>]

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The Laying-Out (infinite abacus)


Consider an abacus with infinitely long columns; this is similar to a num-
ber system with base infinity. (See below for finite approaches.) Now to
add, simply move the requisite number of beads X to the already-calculated
Y; you have X + Y. To subtract, do the reverse. To multiply, three columns
are used. The first column is infinitely deep; the second is set to zero;
the third to Y. Set X on the first; move one bead on the second; set a
second X on the first; move a second bead on the second, and do this until
the second and third columns are equivalent.

One might also use two measure-strings and two columns. Set one measure-
string to X, the second to Y; one column measure out to X, the second set
at 1; measure a second X on the first column, add another 1 on the second
and repeat until the second column is the length of the Y measure-string.

One could use as well just one measure-string and a marker on the side of
column Y; when the beads reach the marker, the calculation is finished.

Division is a reversal of this process; set X and set Y on a measure-
string; subtract the measure-string from X; add one to the second empty
column; repeat until no more subtractings are possible; the length of the
beads in the second column represents the integral quotient; what remains
in the first is the remainder.

The measure-strings in both instances are placed next to the requisite
beads on the first and second columns.

What is unique in this system is that there is no necessity whatsoever to
name the numbers of beads, i.e. assign them to particular integral values.
Instead, one has greater or lesser numbers of beads in the first column;
after the operation, the rough length of the beads is the result. In this
manner, a king may count his horses, a queen her subjects, without further
need of specific tallying. When the column gets especially low, judge as
"more or less"; that is all that is necessary.

This is to some extent the mathematics of the heap or pile, a mathematics
with an inner exactitude, but a fuzzy reading of both givens and results.

Of course one can also consider an abacus with one bead per column and an
infinite number of columns. In this manner, addition and subtraction are
again to the base infinity, simply the moving up and down of beads at the
leading edge of the quantity. But if one is insistent on multiplication, a
second tally is necessary, and if one is insistent on division, one must
look for the same.

Moreover, it is unnecessary to specify an infinite length or number of
beads or positions. An inaccessibly high finite number will do - or even a
finite number practically greater than any conceivable calculations might
warrant. Of course such a number could be arbitrarily extended or retract-
ed by convention or convenience. In any case, problems of platonism or in-
finity are bypassed in this fashion; the systems are both functional and
phenomenologically interesting.

Addition with columns or rows

1oooooooooooooooooooooooooooooo
2oooooooo

1oooooooooooooooooooooooooooooooooooooo
2

Division with columns or rows and tallies (2ooo can be measure-string)

1ooooooooooo
2ooo
3

1oooooooo
2ooo
3o

1ooooo
2ooo
3oo

1oo (remainder)
2ooo
3ooo

I'm fascinated by these simple systems of primitive measurings and tabula-
tions of exactitude, of quantities precisely calculated but unknown, of
the measurings of kingdoms without largesse and the origins of bureaucra-
cies. For nothing more is needed than the laying-out of rows upon the
ground, exalting at the beads disappearing in the distance, and worrying
when the line becomes shorter, almost starved and measurable.


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Appendix of language etiquette:

 oooooooooooooooooooooooooooooo              File: ww
1oooooooooooooooooooooooooooooo

    oooooooo              File: ww
1oooooooooooooooooooooooooooooo
2oooooooo

    oooooooooooooooooooooooooooooooooooooo              File: ww
1oooooooooooooooooooooooooooooooooooooo
 0: oooooooooooooooooooooooooooooo oooooooo
 1: oooooooooooooooooooooooooooooo-oooooooo
 2: oooooooo oooooooooooooooooooooooooooooo
 3: oooooooo-oooooooooooooooooooooooooooooo

    ooo              File: ww
Division with columns or rows and tallies (2ooo can be measure-string)
00: boo 01: coo 02: foo 03: goo 04: loo 05: moo 06: Ofo 07: oho 08: Oto
09: too 10: woo 11: zoo

    ooooooooooo              File: ww
1ooooooooooo
 0: oooooooo ooo
 1: oooooooo-ooo
 2: ooo oooooooo
 3: ooo-oooooooo

   ooooo              File: ww
1ooooo

    oo              File: ww
2ooo
3oo
00: Ao 16: moo 32: Oto 01: bo 17: no 33: ow 02: boo 18: o 34: ox 03: coo
19: od 35: Oz 04: do 20: oe 36: po 05: Fo 21: of 37: Ro 06: foo 22: Ofo
38: so 07: go 23: Og 39: to 08: goo 24: oh 40: too 09: ho 25: oho 41: wo
10: io 26: Ok 42: woo 11: jo 27: om 43: yo 12: ko 28: on 44: zo 13: lo 29:
ooo 45: zoo 14: loo 30: or 15: mo 31: os


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